Incorporation of arterial growth

By reformulation of the SEF with respect to the distortional component of the right Cauchy-Green tensor Cˆ =J^−1/3CasΨ(C) := Ψ( ˆˆ C), it can be shown that the scalar γ can be identified with the hydrostatic pressure p [see e.g. 27]. The general stress-strain relation for an incompressible material hence emerges as

S= 2∂Ψ(C)ˆ

∂C + pJC⁻¹. (2.50)

Near incompressibility Nearly incompressible materials are often used to approximate incompressible material behavior. In the modeling of this sort of materials, the constraint J = 1 is not strictly enforced. Instead the overall SEF is defined as

Ψ(C) = ˆΨ(C) + Ψ_vol(J), (2.51) whereby the volumetric strain energy function Ψ_vol acts as a penalty for volumetric deformations. A very simple choice for the volumetric function is Ψ_vol = ¹₂κ(J −1)². This choice reflects the solution of the variational formulation (2.38) subject to the incompressibility constraint by means of a perturbed Lagrangian formulation [233].

Polyconvexity The existence of solutions of the weak formulation (2.38) depends on some specific properties of the SEF Ψ˜. In fact, it can be shown that the solution of (2.38) corresponds to the minimization of the functional

W(U) = Z

Ω0

Ψ(X,˜ F(X)) dV (2.52) with respect toU [see e.g. 174]. The existence of a minimizer of the functional (2.52) is generally obtained under restrictions of the solution space and the requirement that the SEF be polyconvex. Abridging the introduction of the theory, the concept of polycon-vexity in 3 dimensions can be reduced to the requirement that the SEF can be expressed in terms of a convex function Ψ_pcx via

Ψ(F) = Ψ˜ _pcx(F,cofF,detF). (2.53) A general introduction to this topic can be found in Dacorogna [43]. An analysis of some common nearly incompressible formulations is given by Hartmann and Neff [92].

2.2. Incorporation of arterial growth

The continuum mechanical description of a material introduced in section 2.1.3 material-ized from an elasticity argument. Functional adaption, i.e., the adaption of living tissue to external stimuli [68], clearly is a process which cannot be cast in the framework of elasticity. Thus, the incorporation of such irreversible processes necessitates an extension of the presented continuum mechanical description.

Modern formulations of growth are often referring back to Thompson [216] who opined on a mechanistic view on growth and the inherent connection between growth and form.

The reformulation of this line of thoughts in terms of a kinematic description of growth

is often credited to Skalak et al. [204]. Motivated by this work, Rodriguez et al. [187]

introduced the concept of the multiplicative split of the deformation gradient to allow for a continuum mechanical treatment of growth. The theoretical foundation of this approach is based in the field of finite strain plasticity [134]. The multiplicative split

F=F_gF_e (2.54)

as the decomposition of the deformation gradient into an elastic component F_e and an inelastic/growth component F_g implies the existence of an intermediate configuration Ω_g, see figure 2.3. It is important to highlight that this intermediate configuration need

Ω0, TXΩ0

Ωt, T_ϕ(X)Ωt

F

Fg Fe

Ωg, Tφ(X)Ωg

Figure 2.3.: Illustration of the multiplicative split of the deformation gradient into a growth componentFg and an elastic componentFegiving rise to an intermediate configu-rationΩgthat need not necessarily be compatible (as might erroneously be deduced from the drawing).

not be kinematically compatible to still obtain a compatible spatial configuration. As a consequence, the mapping ϕ_g(X) : Ω₀ → Ω_g is not necessarily differentiable. Thus, F_g cannot be defined via differentiation of ϕ_g. The concept of incompatible configurations is also utilized in the modeling of residual stresses in living tissue [205].

Generally, the continuum mechanical treatment of growth of living tissue is a current topic of research and is still controversially discussed. Whereas there has been some agreement on the applicability of the multiplicative framework [see e.g. 71, 97, 128, 143, 208], it has also been questioned due to its phenomenological approach. Particularly with respect to the accurate modeling of constant mass turnover, i.e., the constant formation and degradation of mass of particular constituents of the tissue, the constrained mixture theory [102] claims superiority at the cost of increased model complexity. Since the focus of the work presented here is on parameter identification and predictive modeling, the reader is referred to Ambrosi et al. [5] for a more detailed review on the current state of growth modeling. Here instead, the incorporation and specification of a growth model based on the multiplicative split (2.54) into the continuum mechanical framework is pursued. A detailed introduction to this approach is also provided by Tinkl [218].

Isotropic growth Isotropic growth is a subclass of the multiplicative formulation ob-tained by specifying growth in terms of the so called growth stretch ϑ(X) ∈ R. The

2.2. Incorporation of arterial growth growth stretch is used to uniformly define the deformation gradient F_g =ϑI such that the overall deformation gradient is given by

F=ϑF_e. (2.55)

Since growth itself is not supposed to induce stresses directly, the SEF is formally refor-mulated with respect to a growth-free deformation according to

Ψ_e(C) := Ψ(C_e), (2.56)

with C_e = F^>_eF_e = _ϑ¹2F^>F = _ϑ¹2C. For a fixed ϑ, the SEF Ψ_e admits the path independent property necessary to allow for the definition according to (2.44). The stress-strain relation resulting from an elastic deformation only is then given by

S_e= 2∂Ψ_e

∂Ce

. (2.57)

And the relation between the second Piola-Kirchhoff stressS and the elastic component S_e is given according to (2.56) by

S_e = 2∂Ψ_e

To close the stress-strain relation, it remains to define the growth stretch ϑ. Again there is no unified theory available, and the application of the multiplicative framework is mainly based on ad-hoc formulae for the evolution of ϑ. These can be motivated by analyzing the change of massm˙ induced by the growth. The massmof the body can be expressed by whereby incompressibility of the elastic deformation, i.e.,J_e= det(F_e) = 1, was assumed.

Using Jg =ϑ³, the change of massm˙ is computed as Thus, with the elastic part of the overall deformation being incompressible, the change in densityρ˙₀ in the reference configuration is driven by the evolution of the stretch ratio ϑ˙, motivating relations of the form

ϑ˙ =f(ϑ, ...). (2.61)

A popular choice is to express the evolution ofϑon some measure of the pressure. E.g., ϑ˙=f(ϑ,C_e:S_e) =k_ϑ(ϑ)C_e:S_e (2.62) is a popular approach [97, 128]. Therein, the functionk_ϑ is defined by

k_ϑ(ϑ) =

with the constants k_ϑ⁺, k_ϑ⁻, ϑ⁺, ϑ⁻, m⁺_ϑ and m⁻_ϑ, which have to be defined on a patient-specific basis. The predictive capabilities of such a model are highly related to the identifiability of the parameters involved. Furthermore, these formulae don’t model the biochemical processes in the living tissue but are merely the result of a top-down modeling approach. Thus, with the goal of identifying parameters, it is reasonable to accumulate the function k_ϑ(ϑ)from (2.63) in a single parameterc_ϑ resulting in

k_ϑ(ϑ) =c_ϑ. (2.64)

The resulting relation (2.62) can be further simplified by ignoring the dependency of the measure of pressure such that the evolution of the growth stretch is directly given by

ϑ˙=c_ϑ. (2.65)

With respect to the accurate modeling of the physical processes of growth, the simple growth law (2.65) is clearly degenerated. However, with regard to predictive modeling in the context of a parameter identification framework, it offers clear advantages. By defining the parameterc_ϑas the parameter to be identified, the identification process has a direct flexible control over the volumetric growth stretchϑ. In contrast to complicated formulations such as (2.63), this increases the flexibility and can therefore have a benefi-cial effect on the nature of the identification problem. Furthermore, the high number of parameters in the formulation (2.63) can have a negative effect on the identifiability of the parameters. Another important advantage of the simple formulation (2.65) is given by its independence of a homeostatic (i.e., healthy) state.